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21  Time Series Analysis

21.1 Getting Started

21.1.1 Load Packages

library("petersenlab")
library("xts")
library("zoo")
library("forecast")
library("brms")
library("plotly")
library("tidyverse")

21.1.2 Load Data

load(file = "./data/player_stats_weekly.RData")
load(file = "./data/player_stats_seasonal.RData")
load(url("https://osf.io/download/q6rjf/"))

21.2 Overview of Time Series Analysis

Time series analysis is useful when trying to generate forecasts from longitudinal data. That is, time series analysis seeks to evaluate change over time to predict future values.

There many different types of time series analyses. For simplicity, in this chapter, we use autoregressive integrated moving average (ARIMA) models to demonstrate one approach to time series analysis. We also leverage Bayesian mixed models to generate forecasts of future performance and plots of individuals model-implied performance by age and position.

21.3 Autoregressive Integrated Moving Average (ARIMA) Models

Hyndman & Athanasopoulos (2021) provide a nice overview of ARIMA models. As noted by Hyndman & Athanasopoulos (2021), ARIMA models aim to describe how a variable is correlated with itself over time (autocorrelation)—i.e., how earlier levels of a variable are correlated with later levels of the same variable. ARIMA models perform best when there is a clear pattern where later values are influenced by earlier values. ARIMA models incorporate autoregression effects, moving average effects, and differencing.

ARIMA models can have various numbers of terms and model complexity. They are specified in the following form: \(\text{ARIMA}(p,d,q)\), where:

• \(p =\) the number of autoregressive terms
• \(d =\) the number of differences between consecutive scores (to make the time series stationary by reducing trends and seasonality)
• \(q =\) the number of moving average terms

ARIMA models assume that the data are stationary (i.e., there are no long-term trends), are non-seasonal (i.e., there is no consistency of the timing of the peaks or troughs in the line), and that earlier values influence later values. This may not strongly be the case in fantasy football, so ARIMA models may not be particularly useful in forecasting fantasy football performance. Other approaches, such as exponential smoothing, may be useful for data that show longer-term trends and seasonality (Hyndman & Athanasopoulos, 2021). Nevertheless, ARIMA models are widely used in forecasting financial markets and economic indicators. Thus, it is a useful technique to learn.

Adapted from: https://rc2e.com/timeseriesanalysis (archived at https://perma.cc/U5P6-2VWC).

21.3.1 Create the Time Series Objects

weeklyFantasyPoints_tomBrady <- player_stats_weekly_offense %>% 
  filter(
    player_id == "00-0019596" | player_display_name == "Tom Brady")

weeklyFantasyPoints_peytonManning <- player_stats_weekly_offense %>% 
  filter(
    player_id == "00-0010346" | player_display_name == "Peyton Manning")

ts_tomBrady <- xts::xts(
  x = weeklyFantasyPoints_tomBrady["fantasy_points"],
  order.by = weeklyFantasyPoints_tomBrady$gameday)

ts_peytonManning <- xts::xts(
  x = weeklyFantasyPoints_peytonManning["fantasy_points"],
  order.by = weeklyFantasyPoints_peytonManning$gameday)

ts_tomBrady

           fantasy_points
2000-11-23           0.24
2001-09-23           2.74
2001-09-30           6.92
2001-10-07           0.34
2001-10-14          22.56
2001-10-21          19.88
2001-10-28           8.02
2001-11-04          22.00
2001-11-11           4.18
2001-11-18           8.00
       ...               
2022-11-06          15.20
2022-11-13          16.02
2022-11-27          18.04
2022-12-05          17.14
2022-12-11          10.12
2022-12-18          16.58
2022-12-25          11.34
2023-01-01          37.68
2023-01-08           7.36
2023-01-16          22.04

ts_peytonManning

           fantasy_points
1999-09-12          15.06
1999-09-19          16.40
1999-09-26          29.56
1999-10-10          19.66
1999-10-17          10.10
1999-10-24          17.86
1999-10-31          17.76
1999-11-07          17.76
1999-11-14          15.18
1999-11-21          22.00
       ...               
2015-10-04           8.32
2015-10-11           6.64
2015-10-18           9.60
2015-11-01          11.60
2015-11-08          15.24
2015-11-15          -6.60
2016-01-03           2.56
2016-01-17          10.78
2016-01-24          14.14
2016-02-07           3.64

ts_combined <- merge(
  ts_tomBrady,
  ts_peytonManning
)

names(ts_combined) <- c("Tom Brady","Peyton Manning")

21.3.2 Plot the Time Series

plot(
  ts_tomBrady,
  main = "Tom Brady's Fantasy Points by Game")

Figure 21.1: Tom Brady’s Historical Fantasy Points by Game.

plot(
  ts_combined,
  legend,
  legend.loc = "topright",
  main = "Fantasy Points by Game")

Figure 21.2: Historical Fantasy Points by Game for Tom Brady and Peyone Manning.

21.3.3 Rolling Mean/Median

zoo::rollmean(
  x = ts_tomBrady,
  k = 5)

           fantasy_points
2001-09-30          6.560
2001-10-07         10.488
2001-10-14         11.544
2001-10-21         14.560
2001-10-28         15.328
2001-11-04         12.416
2001-11-11         13.984
2001-11-18         14.084
2001-11-25         10.568
2001-12-02         11.488
       ...               
2022-10-23         15.492
2022-10-27         14.748
2022-11-06         15.612
2022-11-13         16.700
2022-11-27         15.304
2022-12-05         15.580
2022-12-11         14.644
2022-12-18         18.572
2022-12-25         16.616
2023-01-01         19.000

zoo::rollmedian(
  x = ts_tomBrady,
  k = 5)

           fantasy_points
2001-09-30           2.74
2001-10-07           6.92
2001-10-14           8.02
2001-10-21          19.88
2001-10-28          19.88
2001-11-04           8.02
2001-11-11           8.02
2001-11-18           8.52
2001-11-25           8.00
2001-12-02           8.52
       ...               
2022-10-23          15.20
2022-10-27          15.20
2022-11-06          16.02
2022-11-13          17.10
2022-11-27          16.02
2022-12-05          16.58
2022-12-11          16.58
2022-12-18          16.58
2022-12-25          11.34
2023-01-01          16.58

21.3.4 Autocorrelation

The autocorrelation function (ACF) plot depicts the autocorrelation of scores as a function of the length of the lag. Significant autocorrelation is detected when the autocorrelation exceeds the dashed blue lines, as is depicted in Figure 21.3.

acf(ts_tomBrady)

Figure 21.3: Autocorrelation Function (ACF) Plot of Tom Brady’s Historical Fantasy Points by Game.

Box.test(ts_tomBrady)

    Box-Pierce test

data:  ts_tomBrady
X-squared = 6.9696, df = 1, p-value = 0.008291

21.3.5 Fit an Autoregressive Integrated Moving Average Model

forecast::auto.arima(ts_tomBrady)

Series: ts_tomBrady 
ARIMA(5,1,4) 

Coefficients:
         ar1      ar2      ar3      ar4     ar5      ma1     ma2      ma3
      0.4519  -0.4600  -0.1781  -0.0834  0.2287  -1.3634  0.8837  -0.1480
s.e.  0.2444   0.2272   0.3634   0.0585  0.0535   0.2504  0.4110   0.5295
          ma4
      -0.3333
s.e.   0.3496

sigma^2 = 59.26:  log likelihood = -1318.42
AIC=2656.83   AICc=2657.42   BIC=2696.29

forecast::auto.arima(ts_peytonManning)

Series: ts_peytonManning 
ARIMA(1,0,1) with non-zero mean 

Coefficients:
         ar1      ma1     mean
      0.9134  -0.8192  17.4587
s.e.  0.0605   0.0786   0.9682

sigma^2 = 60.52:  log likelihood = -959.87
AIC=1927.73   AICc=1927.88   BIC=1942.23

arima_tomBrady <- arima(
  ts_tomBrady,
  order = c(5, 1, 4))

summary(arima_tomBrady)

Call:
arima(x = ts_tomBrady, order = c(5, 1, 4))

Coefficients:
         ar1      ar2      ar3      ar4     ar5      ma1     ma2      ma3
      0.4519  -0.4600  -0.1781  -0.0834  0.2287  -1.3634  0.8837  -0.1480
s.e.  0.2444   0.2272   0.3634   0.0585  0.0535   0.2504  0.4110   0.5295
          ma4
      -0.3333
s.e.   0.3496

sigma^2 estimated as 57.86:  log likelihood = -1318.42,  aic = 2656.83

Training set error measures:
                    ME     RMSE      MAE       MPE     MAPE      MASE
Training set 0.6138177 7.596882 6.059765 -26.43736 54.72413 0.7239184
                     ACF1
Training set -0.005875514

confint(arima_tomBrady)

          2.5 %      97.5 %
ar1 -0.02718585  0.93100467
ar2 -0.90534347 -0.01472881
ar3 -0.89026492  0.53413092
ar4 -0.19813198  0.03125533
ar5  0.12376560  0.33358684
ma1 -1.85414989 -0.87259362
ma2  0.07825054  1.68916699
ma3 -1.18577457  0.88978834
ma4 -1.01847540  0.35186954

forecast::checkresiduals(arima_tomBrady)

    Ljung-Box test

data:  Residuals from ARIMA(5,1,4)
Q* = 2.2811, df = 3, p-value = 0.5162

Model df: 9.   Total lags used: 12

Figure 21.4: Model Summary of Autoregressive Integrated Moving Average Model fit to Tom Brady’s Historical Performance by Game.

arima_tomBrady_removeNonSigTerms <- arima(
  ts_tomBrady,
  order = c(5, 1, 4),
  fixed = c(NA, NA, 0, NA, NA, NA, NA, NA, NA))

summary(arima_tomBrady_removeNonSigTerms)

Call:
arima(x = ts_tomBrady, order = c(5, 1, 4), fixed = c(NA, NA, 0, NA, NA, NA, 
    NA, NA, NA))

Coefficients:
         ar1      ar2  ar3      ar4     ar5      ma1     ma2      ma3      ma4
      0.5195  -0.5537    0  -0.0923  0.2248  -1.4320  1.0411  -0.4109  -0.1649
s.e.  0.2244   0.2095    0   0.0579  0.0559   0.2311  0.3713   0.2145   0.0657

sigma^2 estimated as 57.92:  log likelihood = -1318.59,  aic = 2655.18

Training set error measures:
                    ME     RMSE      MAE       MPE     MAPE      MASE
Training set 0.6222971 7.600691 6.078548 -26.34234 54.91081 0.7261622
                     ACF1
Training set -0.005570349

confint(arima_tomBrady_removeNonSigTerms)

          2.5 %       97.5 %
ar1  0.07978268  0.959242224
ar2 -0.96433523 -0.143157528
ar3          NA           NA
ar4 -0.20569255  0.021080469
ar5  0.11530044  0.334344203
ma1 -1.88493886 -0.979076737
ma2  0.31341100  1.768874322
ma3 -0.83134627  0.009456031
ma4 -0.29362802 -0.036192104

forecast::checkresiduals(arima_tomBrady_removeNonSigTerms)

    Ljung-Box test

data:  Residuals from ARIMA(5,1,4)
Q* = 2.3773, df = 3, p-value = 0.4979

Model df: 9.   Total lags used: 12

Figure 21.5: Model Summary of modified Autoregressive Integrated Moving Average Model fit to Tom Brady’s Historical Performance by Game.

arima_peytonManning <- arima(
  ts_peytonManning,
  order = c(1, 0, 1))

summary(arima_peytonManning)

Call:
arima(x = ts_peytonManning, order = c(1, 0, 1))

Coefficients:
         ar1      ma1  intercept
      0.9134  -0.8192    17.4587
s.e.  0.0605   0.0786     0.9682

sigma^2 estimated as 59.86:  log likelihood = -959.87,  aic = 1927.73

Training set error measures:
                      ME     RMSE      MAE       MPE     MAPE      MASE
Training set 0.005206747 7.736947 6.030506 -99.38311 124.2982 0.7309896
                     ACF1
Training set -0.008272646

confint(arima_peytonManning)

               2.5 %     97.5 %
ar1        0.7949304  1.0319427
ma1       -0.9732963 -0.6650994
intercept 15.5611483 19.3563260

forecast::checkresiduals(arima_peytonManning)

    Ljung-Box test

data:  Residuals from ARIMA(1,0,1) with non-zero mean
Q* = 3.8411, df = 8, p-value = 0.8712

Model df: 2.   Total lags used: 10

Figure 21.6: Model Summary of Autoregressive Integrated Moving Average Model fit to Peyton Manning’s Historical Performance by Game.

21.3.6 Generate the Model Forecasts

forecast_tomBrady <- forecast::forecast(
  arima_tomBrady,
  level = c(80, 95)) # 80% and 95% confidence intervals

forecast_peytonManning <- forecast::forecast(
  arima_peytonManning,
  level = c(80, 95)) # 80% and 95% confidence intervals

forecast_tomBrady

    Point Forecast     Lo 80    Hi 80     Lo 95    Hi 95
384       21.03918 11.290652 30.78772 6.1300916 35.94828
385       15.83752  6.050851 25.62418 0.8701029 30.80493
386       22.89837 13.062972 32.73377 7.8564254 37.94032
387       18.55187  8.527204 28.57654 3.2204660 33.88327
388       17.70611  7.680168 27.73206 2.3727526 33.03948
389       18.27131  8.148171 28.39445 2.7893058 33.75331
390       17.91113  7.751759 28.07050 2.3737135 33.44854
391       19.61626  9.454630 29.77790 4.0753848 35.15715
392       19.52851  9.362719 29.69431 3.9812732 35.07575
393       18.52801  8.361647 28.69436 2.9799013 34.07611

forecast_peytonManning

    Point Forecast    Lo 80    Hi 80      Lo 95    Hi 95
278       12.71938 2.804079 22.63467 -2.4447620 27.88351
279       13.12963 3.170404 23.08886 -2.1016934 28.36096
280       13.50437 3.508639 23.50011 -1.7827835 28.79153
281       13.84668 3.820584 23.87277 -1.4869093 29.18027
282       14.15935 4.107997 24.21070 -1.2128686 29.53157
283       14.44496 4.372576 24.51734 -0.9594210 29.84933
284       14.70584 4.615948 24.79573 -0.7253186 30.13700
285       14.94414 4.839661 25.04862 -0.5093279 30.39761
286       15.16181 5.045178 25.27845 -0.3102454 30.63387
287       15.36064 5.233877 25.48741 -0.1269092 30.84819

21.3.7 Plot the Model Forecasts

forecast::autoplot(forecast_tomBrady) + 
  labs(
    x = "Game Number",
    y = "Fantasy Points",
    title = "Tom Brady's Historical and Projected Fantasy Points by Game",
    subtitle = "(if he were to have continued playing additional seasons)"
  ) +
  theme_classic()

Figure 21.7: Tom Brady’s Historical and Projected Fantasy Points by Game.

forecast::autoplot(forecast_peytonManning) + 
  labs(
    x = "Game Number",
    y = "Fantasy Points",
    title = "Peyton Manning's Historical and Projected Fantasy Points by Game",
    subtitle = "(if he were to have continued playing additional seasons)"
  ) +
  theme_classic()

Figure 21.8: Peyton Manning’s Historical and Projected Fantasy Points by Game.

21.4 Bayesian Mixed Models

The Bayesian longitudinal mixed models were estimated in Section 12.3.5.

21.4.1 Prepare New Data Object

player_stats_seasonal_offense_subset <- player_stats_seasonal_offense %>% 
  filter(position_group %in% c("QB","RB","WR","TE"))

player_stats_seasonal_offense_subset$position[which(player_stats_seasonal_offense_subset$position == "HB")] <- "RB"

player_stats_seasonal_kicking_subset <- player_stats_seasonal_kicking %>% 
  filter(position == "K")

player_stats_seasonal_offense_subset <- bind_rows(
  player_stats_seasonal_offense_subset,
  player_stats_seasonal_kicking_subset
)

player_stats_seasonal_offense_subset$player_idFactor <- factor(player_stats_seasonal_offense_subset$player_id)
player_stats_seasonal_offense_subset$positionFactor <- factor(player_stats_seasonal_offense_subset$position)

player_stats_seasonal_offense_subsetCC <- player_stats_seasonal_offense_subset %>%
  filter(
    !is.na(player_idFactor),
    !is.na(fantasy_points),
    !is.na(positionFactor),
    !is.na(ageCentered20),
    !is.na(ageCentered20Quadratic),
    !is.na(years_of_experience))

player_stats_seasonal_offense_subsetCC <- player_stats_seasonal_offense_subsetCC %>% 
  filter(player_id %in% bayesianMixedModelFit$data$player_idFactor) %>% 
  mutate(positionFactor = droplevels(positionFactor))

player_stats_seasonal_offense_subsetCC <- player_stats_seasonal_offense_subsetCC %>%
  group_by(player_id) %>% 
  group_modify(~ add_row(.x, season = max(player_stats_seasonal_offense_subsetCC$season) + 1)) %>% 
  fill(player_display_name, player_idFactor, position, position_group, positionFactor, team, .direction = "downup") %>% 
  ungroup

player_stats_seasonal_offense_subsetCC <- player_stats_seasonal_offense_subsetCC %>% 
  left_join(
    player_stats_seasonal_offense_subsetCC %>% 
      filter(season == max(player_stats_seasonal_offense_subsetCC$season) - 1) %>% 
      select(player_id, age_lastYear = age, years_of_experience_lastYear = years_of_experience), 
    by = "player_id") %>%
  mutate(
    age = if_else(season == max(player_stats_seasonal_offense_subsetCC$season), age_lastYear + 1, age), # increment age by 1
    ageCentered20 = age - 20,
    years_of_experience = if_else(season == max(player_stats_seasonal_offense_subsetCC$season), years_of_experience_lastYear + 1, years_of_experience)) # increment experience by 1

activePlayers <- unique(player_stats_seasonal_offense_subsetCC[c("player_id","season")]) %>% 
  filter(season == max(player_stats_seasonal_offense_subsetCC$season) - 1) %>% 
  select(player_id) %>% 
  pull()

inactivePlayers <- player_stats_seasonal_offense_subsetCC$player_id[which(player_stats_seasonal_offense_subsetCC$player_id %ni% activePlayers)]

player_stats_seasonal_offense_subsetCC <- player_stats_seasonal_offense_subsetCC %>% 
  filter(player_id %in% activePlayers | (player_id %in% inactivePlayers & season < max(player_stats_seasonal_offense_subsetCC$season) - 1)) %>% 
  mutate(
    player_idFactor = droplevels(player_idFactor) 
  )

21.4.2 Generate Predictions

player_stats_seasonal_offense_subsetCC$fantasyPoints_bayesian <- predict(
  bayesianMixedModelFit,
  newdata = player_stats_seasonal_offense_subsetCC
)[,"Estimate"]

21.4.3 Table of Next Season Predictions

player_stats_seasonal_offense_subsetCC %>% 
  filter(season == max(player_stats_seasonal_offense_subsetCC$season), position == "QB") %>%
  arrange(-fantasyPoints_bayesian) %>% 
  select(player_display_name, fantasyPoints_bayesian)

player_stats_seasonal_offense_subsetCC %>% 
  filter(season == max(player_stats_seasonal_offense_subsetCC$season), position == "RB") %>%
  arrange(-fantasyPoints_bayesian) %>% 
  select(player_display_name, fantasyPoints_bayesian)

player_stats_seasonal_offense_subsetCC %>% 
  filter(season == max(player_stats_seasonal_offense_subsetCC$season), position == "WR") %>%
  arrange(-fantasyPoints_bayesian) %>% 
  select(player_display_name, fantasyPoints_bayesian)

player_stats_seasonal_offense_subsetCC %>% 
  filter(season == max(player_stats_seasonal_offense_subsetCC$season), position == "TE") %>%
  arrange(-fantasyPoints_bayesian) %>% 
  select(player_display_name, fantasyPoints_bayesian)

21.4.4 Plot of Individuals’ Model-Implied Predictions

21.4.4.1 Quarterbacks

plot_individualFantasyPointsByAgeQB <- ggplot(
  data = player_stats_seasonal_offense_subsetCC %>% filter(position == "QB"),
  mapping = aes(
    x = round(age, 2),
    y = round(fantasyPoints_bayesian, 2),
    group = player_id)) +
  geom_smooth(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    se = FALSE,
    linewidth = 0.5,
    color = "black") +
  geom_point(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    size = 1,
    color = "transparent" # make points invisible but keep tooltips
  ) +
  labs(
    x = "Player Age (years)",
    y = "Fantasy Points (Season)",
    title = "Fantasy Points (Season) by Player Age: Quarterbacks"
  ) +
  theme_classic()

ggplotly(
  plot_individualFantasyPointsByAgeQB,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)

Figure 21.9: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Quarterbacks.

21.4.4.2 Running Backs

plot_individualFantasyPointsByAgeRB <- ggplot(
  data = player_stats_seasonal_offense_subsetCC %>% filter(position == "RB"),
  mapping = aes(
    x = age,
    y = fantasyPoints_bayesian,
    group = player_id)) +
  geom_smooth(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    se = FALSE,
    linewidth = 0.5,
    color = "black") +
  geom_point(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    size = 1,
    color = "transparent" # make points invisible but keep tooltips
  ) +
  labs(
    x = "Player Age (years)",
    y = "Fantasy Points (Season)",
    title = "Fantasy Points (Season) by Player Age: Running Backs"
  ) +
  theme_classic()

ggplotly(
  plot_individualFantasyPointsByAgeRB,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)

Figure 21.10: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Running Backs.

21.4.4.3 Wide Receivers

plot_individualFantasyPointsByAgeWR <- ggplot(
  data = player_stats_seasonal_offense_subsetCC %>% filter(position == "WR"),
  mapping = aes(
    x = age,
    y = fantasyPoints_bayesian,
    group = player_id)) +
  geom_smooth(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    se = FALSE,
    linewidth = 0.5,
    color = "black") +
  geom_point(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    size = 1,
    color = "transparent" # make points invisible but keep tooltips
  ) +
  labs(
    x = "Player Age (years)",
    y = "Fantasy Points (Season)",
    title = "Fantasy Points (Season) by Player Age: Wide Receivers"
  ) +
  theme_classic()

ggplotly(
  plot_individualFantasyPointsByAgeWR,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)

Figure 21.11: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Wide Receivers.

21.4.4.4 Tight Ends

plot_individualFantasyPointsByAgeTE <- ggplot(
  data = player_stats_seasonal_offense_subsetCC %>% filter(position == "TE"),
  mapping = aes(
    x = age,
    y = fantasyPoints_bayesian,
    group = player_id)) +
  geom_smooth(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    se = FALSE,
    linewidth = 0.5,
    color = "black") +
  geom_point(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    size = 1,
    color = "transparent" # make points invisible but keep tooltips
  ) +
  labs(
    x = "Player Age (years)",
    y = "Fantasy Points (Season)",
    title = "Fantasy Points (Season) by Player Age: Wide Receivers"
  ) +
  theme_classic()

ggplotly(
  plot_individualFantasyPointsByAgeTE,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)

Figure 21.12: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Tight Ends.

21.5 Conclusion

21.6 Session Info

sessionInfo()

R version 4.4.1 (2024-06-14)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 22.04.4 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.20.so;  LAPACK version 3.10.0

locale:
 [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8       
 [4] LC_COLLATE=C.UTF-8     LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8   
 [7] LC_PAPER=C.UTF-8       LC_NAME=C              LC_ADDRESS=C          
[10] LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   

time zone: UTC
tzcode source: system (glibc)

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] lubridate_1.9.3   forcats_1.0.0     stringr_1.5.1     dplyr_1.1.4      
 [5] purrr_1.0.2       readr_2.1.5       tidyr_1.3.1       tibble_3.2.1     
 [9] tidyverse_2.0.0   plotly_4.10.4     ggplot2_3.5.1     brms_2.21.0      
[13] Rcpp_1.0.13       forecast_8.23.0   xts_0.14.0        zoo_1.8-12       
[17] petersenlab_1.0.8

loaded via a namespace (and not attached):
  [1] DBI_1.2.3            mnormt_2.1.1         gridExtra_2.3       
  [4] inline_0.3.19        rlang_1.1.4          magrittr_2.0.3      
  [7] tseries_0.10-57      matrixStats_1.3.0    compiler_4.4.1      
 [10] mgcv_1.9-1           loo_2.8.0            vctrs_0.6.5         
 [13] reshape2_1.4.4       quadprog_1.5-8       pkgconfig_2.0.3     
 [16] fastmap_1.2.0        backports_1.5.0      labeling_0.4.3      
 [19] pbivnorm_0.6.0       utf8_1.2.4           cmdstanr_0.8.1.9000 
 [22] rmarkdown_2.28       tzdb_0.4.0           ps_1.7.7            
 [25] xfun_0.47            jsonlite_1.8.8       psych_2.4.6.26      
 [28] parallel_4.4.1       lavaan_0.6-18        cluster_2.1.6       
 [31] R6_2.5.1             stringi_1.8.4        RColorBrewer_1.1-3  
 [34] StanHeaders_2.32.10  rpart_4.1.23         lmtest_0.9-40       
 [37] estimability_1.5.1   rstan_2.32.6         knitr_1.48          
 [40] base64enc_0.1-3      bayesplot_1.11.1     splines_4.4.1       
 [43] timechange_0.3.0     Matrix_1.7-0         nnet_7.3-19         
 [46] tidyselect_1.2.1     rstudioapi_0.16.0    abind_1.4-5         
 [49] yaml_2.3.10          timeDate_4032.109    codetools_0.2-20    
 [52] processx_3.8.4       curl_5.2.2           pkgbuild_1.4.4      
 [55] lattice_0.22-6       plyr_1.8.9           quantmod_0.4.26     
 [58] withr_3.0.1          bridgesampling_1.1-2 urca_1.3-4          
 [61] posterior_1.6.0      coda_0.19-4.1        evaluate_0.24.0     
 [64] foreign_0.8-86       RcppParallel_5.1.9   pillar_1.9.0        
 [67] tensorA_0.36.2.1     checkmate_2.3.2      stats4_4.4.1        
 [70] distributional_0.4.0 generics_0.1.3       TTR_0.24.4          
 [73] hms_1.1.3            mix_1.0-12           rstantools_2.4.0    
 [76] munsell_0.5.1        scales_1.3.0         xtable_1.8-4        
 [79] glue_1.7.0           lazyeval_0.2.2       emmeans_1.10.4      
 [82] Hmisc_5.1-3          tools_4.4.1          data.table_1.16.0   
 [85] mvtnorm_1.3-0        grid_4.4.1           mitools_2.4         
 [88] crosstalk_1.2.1      QuickJSR_1.3.1       colorspace_2.1-1    
 [91] nlme_3.1-164         fracdiff_1.5-3       htmlTable_2.4.3     
 [94] Formula_1.2-5        cli_3.6.3            fansi_1.0.6         
 [97] viridisLite_0.4.2    Brobdingnag_1.2-9    V8_5.0.0            
[100] gtable_0.3.5         digest_0.6.37        farver_2.1.2        
[103] htmlwidgets_1.6.4    htmltools_0.5.8.1    lifecycle_1.0.4     
[106] httr_1.4.7          

Corston, R., & Colman, A. M. (2000). A crash course in SPSS for Windows. Wiley-Blackwell.

Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts. https://otexts.com/fpp3